Finite Group Databasebeta

← #32989812

Subgroups — #32989812

Order 18688.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988793C211yesmaximal, characteristic
2#22610891C211yesmaximal, characteristic, complemented
3#22610724C211yesmaximal, characteristic
4#22609928C211yesmaximal, characteristic
5C2×#17988337C211yesmaximal, characteristic, complemented
6C2×#17988332C211yesmaximal, characteristic, complemented
7C73×#1296C211yesmaximal, characteristic, complemented
8C22×#11924017—21no—
9C2×#14556591C2211yescharacteristic
10C2×C4×D292C2211yescharacteristic
11#17988793—22no—
12#17988793—22no—
13C2×C4⋊#10596831C2211yescharacteristic
14C2×#14556516—21no—
15C2×#14556511—21no—
16C2×#14556503—21no—
17C2×#14556503—21no—
18C2×C4×#10596831C2211yescharacteristic
19#17988349—21no—
20#17988349—21no—
21#17988349—21no—
22#17988337—22no—
23#17988332—22no—
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26#384×C73C2211yescharacteristic
27C22×Dic146—21no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C2×#11924017—24no—
31C2×#11924017—42no—
32C2×#11924017—42no—
33C22×#10596831—21no—
34C22×#10596831D411yescharacteristic
35C22×#10596831D411yescharacteristic
36#14556591—21no—
37#14556591—21no—
38#14556591—21no—
39#14556591D412yes—
40C4×D292—21no—
41C4×D292—21no—
42C4×D292—21no—
43C4×D292—21no—
44C2×C4⋊Dic73—21no—
45C2×C4⋊Dic73—21no—
46C2×C4⋊Dic73D411yescharacteristic
47C2×#11924005—21no—
48C4⋊#10596831—21no—
49C4⋊#10596831—21no—
50#14556516—24no—
51#14556511—42no—
52#14556503—42no—
53#14556503—42no—
54C4×#10596831—21no—
55C4×#10596831—21no—
56C22×C8×C73—21no—
57C22×C8×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C42×C73C2311yescharacteristic
62C4⋊C8×C73—21no—
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65C4⋊C8×C73D412yes—
66C23×D73—21no—
67C2×Dic146—22no—
68C2×Dic146—41no—
69C2×Dic146—41no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292—21no—
76C2×D292—21no—
77C2×D292—21no—
78C2×D292SD1612yes—
79C22×Dic73—21no—
80C22×Dic73—41no—
81C2×C4×D73—41no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84C2×C4×D73—41no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87#11924017—48no—
88C2×#10596831—21no—
89C2×#10596831—21no—
90C2×#10596831—21no—
91C2×#10596831—21no—
92C2×#10596831—21no—
93C2×#10596831—41no—
94C2×#10596831—41no—
95C2×#10596831—41no—
96C2×#10596831SD1612yescomplemented
97C4⋊Dic73—21no—
98C4⋊Dic73—22no—
99C4⋊Dic73—22no—
100C4⋊Dic73—21no—
101C4⋊Dic73—41no—
102C4⋊Dic73—41no—
103#11924005—41no—
104#11924005—41no—
105C22×C4×C73—41no—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73#4111yescharacteristic
109C2×C8×C73—22no—
110C2×C8×C73—41no—
111C2×C8×C73—41no—
112C2×C8×C73—41no—
113C2×C8×C73—41no—
114C2×C8×C73—41no—
115C4⋊C4×C73—41no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—81no—
120C42×C73C2×D411yescharacteristic
121C42×C73C2×D411yescharacteristic
122C42×C73#4111yescharacteristic
123C42×C73#4111yescharacteristic
124C23×C73#12111yescharacteristic
125C22×D73—22no—
126C22×D73—41no—
127C22×D73—41no—
128C22×D73—41no—
129C22×D73—41no—
130C22×D73—41no—
131C22×D73—41no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—81no—
135C2×C4×C73—81no—
136C2×C4×C73—81no—
137C2×C4×C73#12011yescharacteristic
138C2×C4×C73#12111yescharacteristic
139C2×C4×C73#12111yescharacteristic
140C2×C4×C73#12311yescharacteristic
141C2×C4×C73#12311yescharacteristic
142C2×C4×C73#12411yescharacteristic
143C2×C4×C73C2×SD1611yescharacteristic
144C2×C4×C73C2×SD1611yescharacteristic
145C2×C4×C73#13511yescharacteristic
146C2×C4×C73#13511yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C73#13711yescharacteristic
151C2×Dic73—41no—
152C2×Dic73—42no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—81no—
156C2×Dic73—81no—
157D292—22no—
158D292—22no—
159D292—41no—
160D292—41no—
161C4×D73—81no—
162C4×D73—81no—
163C4×D73—81no—
164C4×D73—81no—
165Dic146—42no—
166Dic146—42no—
167Dic146—81no—
168Dic146—81no—
169C8×C73—42no—
170C8×C73—81no—
171#10596831—22no—
172#10596831—41no—
173#10596831—81no—
174C22×C73#39411yescharacteristic
175C22×C73#46111yescharacteristic
176C22×C73#46311yescharacteristic
177C22×C73#46511yescharacteristic
178C22×C73#46911yescharacteristic
179C22×C73#47111yescharacteristic
180C22×C73#47311yescharacteristic
181C2×D73—42no—
182C2×D73—42no—
183C2×D73—42no—
184C2×D73—81no—
185C2×D73—81no—
186C2×D73—81no—
187C2×D73—81no—
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—161no—
191C4×C73#44911yescharacteristic
192C4×C73#45011yescharacteristic
193C4×C73#46011yescharacteristic
194C4×C73#46411yescharacteristic
195C4×C73#46511yescharacteristic
196C4×C73#46811yescharacteristic
197C4×C73#48111yescharacteristic
198C4×C73#48211yescharacteristic
199Dic73—81no—
200Dic73—82no—
201#10477—731nomaximal, Sylow
202C2×C73#167111yescharacteristic
203C2×C73#167911yescharacteristic
204C2×C73#169311yescharacteristic
205C2×C73#170111yescharacteristic
206C2×C73#203411yescharacteristic
207C2×C73#205211yescharacteristic
208C2×C73#205511yescharacteristic
209D73—82no—
210C2×#461—731no—
211#1990—731no—
212#1954—731no—
213#1802—731no—
214C2×#334—731no—
215C2×#332—731no—
216#1296D7311yescharacteristic, complemented
217C73#1047711yescharacteristic, complemented, Sylow
218C22×SD16—1461no—
219C2×C4⋊Q8—731no—
220C2×C4×D4—731no—
221#461—1462no—
222#461—1462no—
223C2×C4⋊C8—731no—
224C2×C4⋊C8C2×D7311yescharacteristic
225C2×C4⋊C8C2×D7311yescharacteristic
226C2×#103—1461no—
227C2×#102—1461no—
228C2×C22⋊C8—1461no—
229C2×C22⋊C8—1461no—
230C2×C4×C8—731no—
231#384C2×D7311yescharacteristic
232#340—1461no—
233#340—1461no—
234#340—1461no—
235#334—1462no—
236#332—1462no—
237C22×Q8—1461no—
238C22×D4—731no—
239C23×C4—1461no—
240C2×SD16—2922no—
241C2×SD16—2922no—
242C2×SD16—1464no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—731no—
246C22×C8—731no—
247C22×C8—1461no—
248C4⋊Q8—732no—
249C4⋊Q8—1461no—
250C4⋊Q8—1461no—
251C4⋊Q8—1461no—
252C4×D4—1461no—
253C4×D4—1461no—
254C4×D4—1461no—
255C4×D4—1461no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—731no—
260C2×C4⋊C4—1461no—
261C2×C4⋊C4—1461no—
262C2×C22⋊C4—1461no—
263C2×C42C22×D7311yescharacteristic
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—1461no—
268C4⋊C8—1461no—
269C4⋊C8D29212yes—
270#103—1464no—
271#102—2922no—
272C22⋊C8—2922no—
273C22⋊C8—2922no—
274C4×C8—1461no—
275C4×C8—1461no—
276C24—1461no—
277C2×Q8—2921no—
278C2×Q8—2921no—
279C2×Q8—2921no—
280C2×Q8—2921no—
281C2×Q8—2921no—
282C2×Q8—1462no—
283C2×D4—732no—
284C2×D4—1461no—
285C2×D4—1461no—
286C2×D4—1461no—
287C2×D4—1461no—
288C2×D4—1461no—
289C22×C4—41no—
290C22×C4—2921no—
291C22×C4—2921no—
292C22×C4—2921no—
293C22×C4—2921no—
294C22×C4—2921no—
295C22×C4—2921no—
296C22×C4—2921no—
297C22×C4—1461no—
298C22×C4D4×D7311yescharacteristic
299C22×C4D4×D7311yescharacteristic
300C22×C4#1192402811yescharacteristic
301SD16—2928no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—2921no—
309C2×C8—2921no—
310C2×C8—2921no—
311C2×C8—732no—
312C2×C8—1461no—
313C2×C8—1461no—
314C2×C8—1461no—
315C2×C8—1461no—
316C2×C8—1461no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—81no—
322C4⋊C4—2921no—
323C4⋊C4—2921no—
324C4⋊C4—1461no—
325C4⋊C4—1462no—
326C4⋊C4—1461no—
327C4⋊C4—1462no—
328C22⋊C4—2921no—
329C22⋊C4—2921no—
330C42D4×D7311yescharacteristic
331C42C2×D29211yescharacteristic
332C42#1192402811yescharacteristic
333C42#1192402911yescharacteristic
334C23—2921no—
335C23—2921no—
336C23—2921no—
337C23—2921no—
338C23—2921no—
339C23—2921no—
340C23—1462no—
341C23#1455656411yescharacteristic
342Q8—2922no—
343Q8—2922no—
344Q8—5841no—
345Q8—5841no—
346D4—2921no—
347D4—2921no—
348D4—1462no—
349D4—1462no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—2921no—
356C2×C4—2922no—
357C2×C4—2921no—
358C2×C4—2921no—
359C2×C4—5841no—
360C2×C4—5841no—
361C2×C4—5841no—
362C2×C4—5841no—
363C2×C4—5841no—
364C2×C4—5841no—
365C2×C4#1455656111yescharacteristic
366C2×C4#1455656411yescharacteristic
367C2×C4#1455656611yescharacteristic
368C2×C4#1455657611yescharacteristic
369C2×C4#1455657611yescharacteristic
370C2×C4#1455657911yescharacteristic
371C2×C4SD16×D7311yescharacteristic
372C2×C4SD16×D7311yescharacteristic
373C2×C4#1455661211yescharacteristic
374C2×C4#1455661411yescharacteristic
375C2×C4#1455661511yescharacteristic
376C2×C4#1455661511yescharacteristic
377C2×C4#1455661811yescharacteristic
378C2×C4#1455662311yescharacteristic
379C8—42no—
380C8—81no—
381C8—2921no—
382C8—5841no—
383C8—1462no—
384C22—2922no—
385C22—2922no—
386C22—2922no—
387C22—5841no—
388C22—5841no—
389C22—5841no—
390C22—5841no—
391C22#1798850511yescharacteristic
392C22#1798879111yescharacteristic
393C22#1798880111yescharacteristic
394C22#1798881911yescharacteristic
395C22#1798883911yescharacteristic
396C22#1798884911yescharacteristic
397C22#1798886111yescharacteristic
398C4—21no—
399C4—21no—
400C4—161no—
401C4—5841no—
402C4—5842no—
403C4#1798873511yescharacteristic
404C4#1798873911yescharacteristic
405C4#1798878611yescharacteristic
406C4#1798881211yescharacteristic
407C4#1798881711yescharacteristic
408C4#1798883411yescharacteristic
409C4#1798890711yescharacteristic
410C4#1798891411yescharacteristic
411C2—5842no—
412C2#2260930911yescharacteristic
413C2#2260935311yescharacteristic
414C2#2260944311yescharacteristic
415C2#2260949111yescharacteristic
416C2#2261112311yescharacteristic
417C2#2261123311yescharacteristic
418C2#2261125011yescharacteristic
subgroupG/H
centerC23#14556564
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1296C2
socleC23×C73#121
prime radicalC22×D292D4
hypercenter#1296D73
hypocenterC73#10477
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22610891C2semidirect
5. C2×#17988337C2semidirect
6. C2×#17988332C2semidirect
7. C73×#1296C2semidirect
96. C2×#10596831SD16semidirect
216. #1296D73semidirect
217. C73#10477semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179887932
#226108912
#226107242
#226099282
C2×#179883372
C2×#179883322
C73×#12962
#1047773